Advanced Method for Small-Signal Stability Assessment based on Neuronal Networks

Transcription

1 1 Advanced Method for Small-Signal Stability Aement baed on Neuronal Networ S. P. Teeuwen, I. Erlich, Member, IEEE, and M. A. El-Sharawi, Fellow, IEEE Abtract-- Thi paper deal with a new method for eigenvalue prediction of critical tability mode of power ytem baed on neural networ. The pecial interet of thi paper i on interarea ocillation of large-cale interconnected power ytem. The exiting method for eigenvalue computation are timeconuming and reuire the entire ytem model that include an extenive number of tate variable. After reducing the input pace and proper training of the neural networ, the tability condition of the power ytem can be predicted with high accuracy. Furthermore, the new advanced method allow to predict a mall variable wide band for the real part of eigenvalue. Index Term Eigenvalue, Inter-Area Ocillation, Neural Networ, Small-Signal Stability I. INTRODUCTION Inter-area ocillation in large-cale power ytem are becoming more common epecially for the European Interconnected Power Sytem UCTE/CENTREL. The ytem ha grown very fat in a hort period of time due to the recent eatward expanion. Thi extenive interconnection alter the tability region of the networ, and the ytem experience inter-area ocillation aociated with the winging of many machine in one part of the ytem againt machine in other part. Moreover, for certain load flow condition, the ytem damping can widely change [1-]. With the deregulation of the electricity maret in Europe, the utilitie are allowed to ell their generated power outide their traditional border and compete directly for cutomer. For economical reaon, the operator are often forced to teer the ytem cloer to the tability limit. Thu, the operator need different computational tool for predicting ytem tability. Thee tool mut be accurate and fat to allow online tability aement. The computation of the mall ignal tability i a time conuming proce for large networ, which include the load flow computation, the linearization at the operating point, and the eigenvalue computation [3]. An alternative method i to ue a neural networ () trained with off-line data for different load flow and ytem condition. By uing, a fat computation of the eigenvalue i poible, provided that the networ i properly deigned and trained. Recent approache howed highly accurate reult regarding the eigenvalue prediction of large-cale power ytem [4]. However, the are trained only for narrow operating condition of the power ytem. In fact, when the operating point of a given power ytem change, the correponding eigenvalue may hift ignificantly. Small change in the networ topology, the load flow, or the voltage level may change the eigenvalue within the wealy damped area too. The introduced in [4] predict the eigenvalue themelve. But a that i able to manage all ytem variation mut be deigned for a variable number of eigenvalue. Hence, the mut have a variable tructure, which i a difficult ta. To addre thi iue, the aement method propoed in thi paper i baed not on the prediction of eigenvalue, but on the prediction of region with one or more dominant eigenvalue of inufficient damping. To generate the training data for the, an obervation area i defined within the complex eigenvalue pace. Thi area i located at a low damping level where inter-area eigenvalue typically occur. Then, the entire obervation area i ampled. The ditance between the ample point and the eigenvalue are computed and the ample point are activated depending on the exitence of dominant eigenvalue. The cloer the eigenvalue to a ample point the higher the activation. Once thi i computed for the entire et of pattern, the i trained with thee activation. After proper training, the can be ued in real time to compute the ample activation. Then, the activation are tranformed into area range where the eigenvalue are located. Thee area are characterized by activation higher than a given limit. II. 16-MACHINE DYNAMIC TEST SYSTEM The PST16 Sytem ued in thi tudy i a dynamic 16- machine tet networ. The main focu i on modeling of power ytem dynamic and inter-area ocillation in the time range of a few econd to minute. The PST16 Sytem i modeled by uing real data from generator and controller for different unit uch a hydro,

2 nuclear and thermal power plant. The ytem conit of 3 trongly mehed area, each having 5-6 generator. Thee area are connected partly by wea tranmiion line. Thu, inter-area ocillation can occur. The ytem i deigned for power exchange between thee area. The firt area (Area A) contain motly hydro power plant and i conidered to be a power exporting area. Area B and area C are load demanding area that import power from area A. The total generation of the ytem i about 16 GW. The one-line diagram of the PST16 Sytem i hown in Figure 1. area i changed, the change i ditributed eually among all the generator in thi area. Furthermore, if the generated power of a particular unit fall below a minimum et value, the unit i diconnected. The model alo allow additional generating unit to be connected when the demand increae. The different load flow cenario reult in generating 3,96 pattern for training. The dominant eigenvalue for all cae are hown in Figure. The lant line in the figure are for contant damping at 0% to 0%. A een in the figure, mot of the cae are for well-damped condition, but in ome cae the eigenvalue hift to the low damping region and can caue ytem intability. Figure 1: One-Line Diagram of the PST 16-Machine Tet Sytem III. FEATURE SELECTION Large-cale power ytem include many feature uch a tranmiion line power flow, generated power and demand. For a large ytem, the feature et i too large for any effective training [5]. Therefore, feature extraction or election techniue mut be ued. In thi tudy, a new multiple tep election method (MSS) i propoed. Thi techniue i baed on the principal component analyi (PCA) and the -Mean cluter algorithm. The PCA techniue i characterized by a high reduction rate and a minimal lo of information. However, the calculation of principal component (PC) reuire all original phyical feature, which i not poible to provide in a largecale power ytem. Therefore, for the propoed application only election techniue are uitable, which are characterized by retaining the mot important original feature vector. However, in thi tudy PCA i utilized for reduction of the dimenionality of feature vector a hown below. Let F be a feature matrix of dimenion p n where n i the number of the original feature vector and p i the number of pattern. The empirical covariance matrix C of the normalized F i computed by C = 1 T F F (1) p 1 Let T be a n n matrix of the eigenvector of C. The diagonal variance matrix Σ i then given by Σ = T T C T () Σ include the variance x. Notice that the eigenvalue λ of the covariance matrix C are eual to the element of the variance matrix Σ. The tandard deviation i alo the ingular value of F: Figure : Computed Eigenvalue of the PST 16-Machine Tet Sytem for 3,96 different Load Flow Condition To generate training data for the, different load flow condition are conidered. When the total generation in one = λ ( 1,,..., n ) (3) = The n eigenvalue of C can be determined and orted in decending order λ λ K λ. While T i an n- 1 n dimenional matrix whoe column are the eigenvector of C,

3 3 T i a n matrix including eigenvector of C correponding to the larget eigenvalue of C. The value of i determined baed on the retained variability of the feature, which i the ratio between the firt eigenvalue and the um of all n eigenvalue [6]. The number ha to be choen o that υ cover nearly the total ignal variability, i.e. υ ha a value near to 1.0. Power Voltage Reduction of Dimenion by PCA & Selection by Clutering Voltage Angle υ i= 1 = n i= 1 λ λ i i (4) + From the baic euation of the PC tranformation follow where ( PC ) F F ( PC ) = F T (5) F (6) ( PC ) T F T contain the elected PC feature vector. According to Euation (6) T can be interpreted a Loading Matrix, and ( PC ) F a the non-normalized Factor Matrix. Now, the idea i to ue the column of T T intead of the high dimenional original feature vector F for clutering. In fact, the column of T T are euivalent to thoe of F but with a much lower dimenion. Therefore, the ued -Mean clutering algorithm [7] how with T T a better performance and accuracy than with the original feature directly. Thi i why both techniue, principal component analyi and clutering, are ued in combination. Neverthele, it i even poible to ip the PCA computation altogether, which mean that the original feature are clutered directly from the beginning. Thi approach i preferable for feature with maller pattern number. The -mean clutering algorithm form group with imilar characteritic [7]. The criterion for the bet grouping i the minimization of the within cluter um of ditance to the cluter centroid. Feature within a cluter how high correlation. On the other hand, independent feature are located alway in different cluter. The -mean algorithm reuire the number of cluter from the beginning. Therefore, to form homogeneou cluter with proper within cluter correlation, ome iteration varying cluter number are neceary. However, the number of cluter p ha to be alway greater than the number of PC to obtain the ame variability a the PC. Becaue of the imilarity between the feature within a cluter, one of them can be elected and the ret can be treated a redundant information. The feature in one cluter, which i cloet to the centroid of thi cluter, will be choen. Thu, a group of p feature will be maintained. Reduction of Dimenion by PCA & Selection by Clutering 50 Input Feature Figure 3: Applied MSS Feature Selection Method In MSS, the proce of electing feature i repeated for extremely inhomogeneou feature. In thi tudy, the original feature et wa eparated into 3 homogeneou ubet including power feature (real power, reactive power), voltage and the correponding voltage angle feature. Then, the feature election method i applied to each ubet. The elected feature are then combined to form a new group, from which the final feature are elected applying the ame method again. Finally, 50 of the original feature remain a input variable. The MSS feature election proce i hown in Figure 3. Although the feature clutering ha been carried out baed on the tranformed feature reduced dimenionality, for training a decribed in the next chapter alway original feature were ued. IV. OBSERVATION AREA The propoed tability aement method reuire that the obervation area in the complex eigenvalue pace i defined firt. The obervation area i located at the region of inufficient damping, where typical inter-area eigenvalue can be found. In thi tudy, the damping range i choen between 4% and 1.5%. Then ample are generated along the real axi ( ) for damping coefficient between 4% and 1.5%. Thi proce i repeated for 4 different freuencie f. The width of and f ampling tep are and f, repectively. Thee tep width are contant for the entire ampling proce at: = f = 0.14 Hz The obervation area of inufficient damping i hown in Figure 4. The ample point are mared by circle.

4 4 4% 3% % 1% 0% -1% -% A B C D f f = (9) 3 Notice that thi choice lead to a ditance of 1 repective 3 between neighboring ample point in and f direction, repectively. The choice of thee parameter i baed on experience with training. Thi caling put emphai on eigenvalue movement along ample point row becaue the are trained by ample point row. Moreover, the imum ditance poible between an eigenvalue and the cloet ample point occur when the eigenvalue i located exactly in the geometrical center of 4 adjacent ample point. According to Euation (7) to (9), the imum ditance can be computed a Figure 4: Computed Eigenvalue of the PST 16-Machine Tet Sytem for 3,96 different Load Flow Condition. The Figure how Sample Point in the Obervation Area The ampling method decribed in Figure 4 produced a et of 4 ample point. To increae the accuracy of the, the obervation area i divided into maller obervation ection (row of ample point) whereby each row of ample point ue it own. Hence, 4 independent are ued. The 4 oberved ection covered by the 4 are defined in Table I. TABLE I SAMPLED OBSERVATION AREA DIVIDED INTO 4 SECTIONS FOR TRAINING Output min / / f / Hz A B C D After the obervation area i ampled, the ditance between the ample point and the eigenvalue are computed. Then, the ditance are ued to compute activation value of the ample point. The complex plane of the eigenvalue i defined by the real component and the freuency. The ample point are defined by their location in thi complex plane (, ). The Euclidean ditance between a given f eigenvalue and a given ample point i computed a follow: d = ev ev f f ev + (7) f Becaue and f ue different unit and cannot be compared directly, both are caled by and. Thee contant are choen in relation to the tep ize in and f direction. Thu, f f ev = (8) d f = + = 1 f 3 (10) Baed on thi imum ditance, the activation value a for a ample i defined a a linear relationhip depending on the ditance by Euation (7) a d = d 0 0 d d d > d (11) The activation a i i computed for a given et of ample point with repect to all eigenvalue of one pattern. The final activation value act for the given ample point i the ummation of all activation a i act (1) = n a i i= 1 where n i the number of conidered eigenvalue. The imum ditance in Euation (8) and the activation function in Euation (9) lead to the minimum activation for a ample point located near an eigenvalue act 0.5 (13) V. EIGENVALUE PREDICTION BY NEURAL NETWORKS The activation of the ample point are computed for all pattern. The data are then normalized and huffled randomly before the 4 are trained independently. In thi tudy, the number of training pattern i,966 and the number of teting pattern i 330. Once the are trained, the output (activation value) i ued to identify the location of the eigenvalue. For all pattern, the activation value given from the ample point are ued to et up an activation diagram, which i ued to contruct a range for the eigenvalue poition a hown in Figure 5 for one pattern. According to Table I, the oberved ection of (A) i within = 0. and = 0.. The activation min diagram i extended outide of thee border by tep, where the activation value of the lat point are ept contant.

5 5 The freuency of the imaginary part i f = 0.66 Hz and i bounded by f f ±. The activation of the ample point in Figure 5(a) are plotted in bold face and connected by line. The imum activation limit (margin) i et at 0.5, which i the minimum activation value according to Euation (11). The interection of the activation border with the activation level curve determine the range r of the real part. The range r and the limit of f form a rectangle repreenting the prediction region. The function of the i to predict the eigenvalue of thi region. In the example given in Figure 5(b), the poition of the eigenvalue i = and f = Hz. The identified eigenvalue i mared with a cro and matche the predicted region exactly. If an eigenvalue i located inide a region, the prediction i conidered correct. If an eigenvalue i located outide the region the error i called fale dimial. If a range i identified, but no eigenvalue i located inide the obervation area, the error i called fale alarm. For each of the 4, the fale dimial and the fale alarm error i computed a given in Euation (14) number of fale dimial or fale alarm E [%] = 100 (14) number of pattern The reult are given in Table II. TABLE II FALSE DISMISSAL AND FALSE ALARM ERRORS OF f (a) r Activation Limit min Fale Dimial Fale Alarm Training Teting Training Teting A 0.00 % 0.00 % 0.00 % 0.00 % B 0.00 % 0.00 % 0.00 % 0.00 % C 0.03 % 0.00 % 0.00 % 0.00 % D 0.00 % 0.00 % 0.00 % 0.00 % The error hown in Table II are almot alway zero. Fale alarm did not occur and fale dimial occur only once for one. In other word, the actual eigenvalue poition i alway within the predicted region. However, it wa oberved that when the range i mall, the accuracy of the i reduced. To addre thi iue, a relative range c i defined a the ratio of the abolute range r to the tep width c = r (15) Table III how the mean and the tandard deviation of c for all contructed range r. TABLE III MEAN AND STANDARD DEVIATION OF ERRORS WHEN THE RELATIVE RANGE ACCORDING EQ. (15) IS USED r Training Teting Mean STD Mean STD A B / / / / C D (b) Figure 5: (a) Activation Diagram for one Pattern, (b) Complex Space with Correponding Eigenvalue Identified Another way for evaluating the accuracy of the i to chec the range with repect to the damping coefficient. For thi purpoe, the real part of r i tranformed into a damping range ξ a hown in Figure 6. The mean and tandard deviation of the damping range are hown in Table IV.

6 6 VII. REFERENCES [1] U. Bachmann, I. Erlich and E. Grebe, Analyi of interarea ocillation in the European electric power ytem in ynchronou parallel operation with the Central-European networ, IEEE PowerTech, Budapet 1999 [] H. Breulmann, E. Grebe, M. Löing, W. Winter, R. Witzmann, P. Dupui, M.P. Houry, T. Margotin, J. Zerényi, J. Dudzi, J. Machowi, L. Martín, J.M. Rodríguez, E. Urretavizcaya, Analyi and Damping of Inter-Area Ocillation in the UCTE/CENTREL Power Sytem, CIGRE , Seion 000 [3] P. Kundur, Power Sytem Stability and Control, McGraw-Hill, New Yor, 1994 [4] S.P. Teeuwen, A. Ficher, I. Erlich, M.A. El-Sharawi, Aement of the Small Signal Stability of the European Interconnected Electric Power Sytem Uing Neural Networ, LESCOPE 001, Halifax, Canada, June 001 [5] S.P. Teeuwen, I. Erlich, M.A. El-Sharawi, Feature Reduction for Neural Networ baed Small-Signal Stability Aement, PSCC 00, Sevilla, Spain, June 00 [6] I.T. Jolliffe, Principal Component Analyi, Springer-Verlag, New Yor, 1986 [7] Viual Numeric: IMSL Fortran 90 MP Library Help, Stat/Library, Volume, Chapter 11: Cluter Analyi Figure 6: Contruction of the Damping Range for the Predicted Eigenvalue TABLE IV ERRORS USING THE DAMPING RANGE ξ Training Teting Mean STD Mean STD A 1.3 % 0.3 % 1.3 % 0.3 % B / / / / C 1.9 % 0.4 % 1.9 % 0.4 % D 3.5 % 0.7 % 3.4 % 0.7 % VI. CONCLUSION Thi paper introduce a new method of eigenvalue prediction for mall-ignal tability aement. The method i flexible in term of networ condition and variable number of eigenvalue. The error, lited in Table II, are almot alway zero. In other word, by the propoed method, the eigenvalue are located within the predicted region. By decreaing the ample tep width, the accuracy can be improved even more, but the time for training increae. For high and medium freuencie ( A to C), the damping of eigenvalue can be predicted in a range, which i maller than 1.9%. Only for low freuencie, the average width of the damping range increae to 3.5%. The reaon i the nonlinear relationhip that exit between the real part and the damping coefficient. The lower the freuency, the larger the damping difference for a contant interval. The propoed method allow the fat aement of mallignal tability for large power ytem baed on a retricted number of proce variable. It i o fat that it can be implemented a an on-line tool. A further advantage of thi method i that only the elected variable mut be exchanged and tranmitted between the different part and owner of the ytem. VIII. BIOGRAPHIES Simon P. Teeuwen (1976) i preently PhD tudent in the Department of Electrical Power Sytem at the Univerity of Duiburg/Germany. He tarted hi tudie at the Univerity of Duiburg in In 000, he went a exchange tudent to the Unverity of Wahington, Seattle, where he performed hi Diploma Thei. After hi return to Germany in 001, he received hi Dipl.-Ing. degree at the Univerity of Duiburg. He i a member of VDE and VDI. Itvan Erlich (1953) received hi Dipl.-Ing. degree in electrical engineering from the Univerity of Dreden/Germany in After hi tudie, he wored in Hungary in the field of electrical ditribution networ. From 1979 to 1991, he joined the Department of Electrical Power Sytem of the Univerity of Dreden again, where he received hi PhD degree in In the period of 1991 to 1998, he wored with the conulting company EAB in Berlin and the Fraunhofer Intitute IITB Dreden repectively. During thi time, he alo had a teaching aignment at the Univerity of Dreden. Since 1998, he i Profeor and head of the Intitute of Electrical Power Sytem at the Univerity of Duiburg/Germany. Hi major cientific interet i focued on power ytem tability and control, modelling and imulation of power ytem dynamic including intelligent ytem application. He i a member of VDE and IEEE. Mohammed A. El-Sharawi received the B.Sc. degree in electrical engineering in 1971 from Cairo High Intitute of Technology, Egypt, and the M.A.Sc. and Ph.D. degree in electrical engineering from the Univerity of Britih Columbia, Vancouver, B.C., Canada, in 1977 and 1980, repectively. In 1980, he joined the Univerity of Wahington, Seattle, a a Faculty Member. He erved a the Chairman of Graduate Studie and Reearch and i preently a Profeor of Electrical Engineering. He i the Vice Preident for Technical Activitie of the Neural Networ Society. He coedited an IEEE tutorial boo on the application of neural networ to power ytem. He organized and taught everal international tutorial on intelligent ytem application, power uality and power ytem, and he organized and chaired numerou panel and pecial eion in IEEE and other international conference. He publihed over 10 paper and boo chapter in thee area and hold even licened patent. He i a member of the editorial board and Aociate Editor of everal journal, including the IEEE TRANSACTIONS ON NEURAL NETWORKS and Engineering Intelligent Sytem.